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Nonlinear vibration control of a cantilevered fluid-conveying pipe using the idea of nonlinear energy sink

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Abstract

Different from the both-end supported pipe conveying fluid as a conservative system, the cantilevered fluid-transporting pipe is a non-conservative system and its dynamic behavior is more complex with flutter instabilities when the flow velocity is beyond the critical value. Indeed, controlling such a flutter system is always challenging in engineering applications. This study presents nonlinear vibrations of cantilevered pipe conveying fluid passively controlled via a nonlinear energy sink (NES). Based on the Hamilton principle, the nonlinear dynamic equations coupling with the NES are derived and discretized using high-order Galerkin method. It is indicated that increasing the mass and damping of NES results in an increase in critical flow velocity. Importantly, the optimal placed position of NES where the critical flow velocity is highest has a strong relationship with the pipe’s flutter mode. In the following, the nonlinear analysis shows the dynamic controlling effect on vibration amplitude of the pipe can be classified to three suppression regions with increasing the flow velocity. Varying the mass, damping and stiffness of NES is followed by variations of the suppression regions which are associated with controlling effects and dynamic behaviors of the pipe system.

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Acknowledgements

The authors acknowledge the support provided by Science and Technology on Reactor System Design Technology Laboratory, Fundamental Research Funds for the Central Universities, HUST (2017KFYXJJ135), Natural Science Foundation of Hubei Province (2017CFB429) and National Natural Science Foundation of China (Nos. 11602090, 11672115 and 11872060).

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Authors and Affiliations

  1. Department of Mechanics, Huazhong University of Science and Technology, Wuhan, 430074, China

    K. Zhou, H. L. Dai, H. Yan, L. Wang & Q. Ni

  2. Hubei Key Laboratory for Engineering Structural Analysis and Safety Assessment, Wuhan, 430074, China

    K. Zhou, H. L. Dai, H. Yan, L. Wang & Q. Ni

  3. Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China, Chengdu, 610213, China

    F. R. Xiong & N. B. Jiang

Authors
  1. K. Zhou
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  2. F. R. Xiong
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  3. N. B. Jiang
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  4. H. L. Dai
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  5. H. Yan
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  6. L. Wang
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  7. Q. Ni
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Correspondence to H. L. Dai.

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All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards.

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Appendix

Appendix

In this part, we offer the elements of matrices written in Sect. 3 in detail and also introduce a kind of algebraic relationship to simplify the derived nonlinear damping and stiffness matrices. Firstly, through some further manipulations, the mass, linear damping, nonlinear damping, linear stiffness and nonlinear stiffness matrices can be written as follows:

$$\begin{aligned} M= & {} \int _0^1 {\varvec{\varphi }}^\mathrm{T}{{\varvec{\varphi }}\mathrm{d}\xi } ,\nonumber \\ C_L= & {} \alpha \int _0^1 {{\varvec{\varphi }}^\mathrm{T}{{{{{{\varvec{\varphi }}}}''''}}}\mathrm{d}\xi } \nonumber \\&+\, 2u\sqrt{\beta }\int _0^1 {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi }} }'}\mathrm{d}\xi } \nonumber \\ K_L= & {} \int _0^1 {{\varvec{\varphi }}^\mathrm{T}{{{{{{\varvec{\varphi }} }''''}}}}\mathrm{d}\xi }\nonumber \\&+\, {u}^{2}\int _0^1 {{\varvec{\varphi }}^\mathrm{T}{{{{\varvec{\varphi }} }''}}\mathrm{d}\xi } \nonumber \\ C_N= & {} \int _0^1 {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi }} }'{\varvec{q}}{\dot{{\varvec{q}}}}}^\mathrm{T}\left( {\int _0^\xi {{\varvec{\varphi }'}^\mathrm{T}{\varvec{\varphi }'}\mathrm{d}\xi } } \right) \mathrm{d}\xi } \nonumber \\&-\, 2u\sqrt{\beta }\int _0^1 {{\varvec{\varphi }}^\mathrm{T}{\varvec{\varphi }'{\varvec{qq}}}^\mathrm{T}\left( {\int _0^\xi {{\varvec{\varphi }'}^\mathrm{T}{{{\varvec{\varphi } }''}}\mathrm{d}\xi } } \right) \mathrm{d}\xi } \nonumber \\&-\, \int _0^1 {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi } }''}{\varvec{q}}{\dot{{\varvec{q}}}}}^\mathrm{T}\left( {\int _\xi ^1 {\int _0^\xi {{\varvec{\varphi }'}^\mathrm{T}{\varvec{\varphi }'}\mathrm{d}\xi } } \mathrm{d}\xi } \right) \mathrm{d}\xi } \nonumber \\&+\, 2u\sqrt{\beta }\int _0^1 {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi } }''}{\varvec{qq}}}^\mathrm{T}\left( {\int _\xi ^1 {\int _0^\xi {{\varvec{\varphi }'}^\mathrm{T}{{{\varvec{\varphi } }''}}\mathrm{d}\xi } } \mathrm{d}\xi } \right) \mathrm{d}\xi } \nonumber \\&-\, 2u\sqrt{\beta }\int _0^1 {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi } }''}{\varvec{qq}}}^\mathrm{T}\left( {\int _\xi ^1 {{\varvec{\varphi }'}^\mathrm{T}{\varvec{\varphi }'}\mathrm{d}\xi } } \right) \mathrm{d}\xi } \nonumber \\ K_N= & {} 2u\sqrt{\beta }\int _0^1 {{\varvec{\varphi }}^\mathrm{T}{\varvec{\varphi }'{\dot{{\varvec{q}}}}{\varvec{q}}}^\mathrm{T}{\varvec{\varphi }'}^\mathrm{T}{\varvec{\varphi }'}\mathrm{d}\xi }\nonumber \\&+\,{u}^{2}\int _0^1 {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi } }''}{\varvec{qq}}}^\mathrm{T}{\varvec{\varphi }'}^\mathrm{T}{\varvec{\varphi }'}\mathrm{d}\xi } \nonumber \\&+\,3\int _0^1 {{\varvec{\varphi }}^\mathrm{T}{\varvec{\varphi }'{\varvec{qq}}}^\mathrm{T}{{{\varvec{\varphi } }''}}^\mathrm{T}{{{{{\varvec{\varphi }} }'''}}}\mathrm{d}\xi } \nonumber \\&+\,\int _0^1 {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi } }''}{\varvec{qq}}}^\mathrm{T}{{{\varvec{\varphi } }''}}^\mathrm{T}{{{\varvec{\varphi } }''}}\mathrm{d}\xi } \nonumber \\&-\, u^{2}\int _0^1 {{\varvec{\varphi }}^\mathrm{T}{\varvec{\varphi }'{\varvec{qq}}}^\mathrm{T}\left( {\int _0^\xi {{\varvec{\varphi }'}^\mathrm{T}{{{{{\varvec{\varphi }} }'''}}}\mathrm{d}\xi } } \right) \mathrm{d}\xi } \nonumber \\&+\, \int _0^1 {{\varvec{\varphi }}^\mathrm{T}{\varvec{\varphi }'{\varvec{qq}}}^\mathrm{T}\left( {\int _0^\xi {{{{\varvec{\varphi } }''}}^\mathrm{T}{{{{{{\varvec{\varphi '}}}}'''}}}\mathrm{d}\xi } } \right) \mathrm{d}\xi } \nonumber \\&+\,\, u^{2}\int _0^1 {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi } }''}{\varvec{qq}}}^\mathrm{T}\left( {\int _\xi ^1 {\int _0^\xi {{\varvec{\varphi }'}^\mathrm{T}{ {{{{\varvec{\varphi }} }'''}}}\mathrm{d}\xi } } \mathrm{d}\xi } \right) \mathrm{d}\xi } \nonumber \\&-\, \int _0^1 {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi } }''}{\varvec{qq}}}^\mathrm{T}\left( {\int _\xi ^1 {\int _0^\xi {{{{\varvec{\varphi } }''}}^\mathrm{T}{{{{{{\varvec{\varphi }} }''''}}}}\mathrm{d}\xi } } \mathrm{d}\xi } \right) \mathrm{d}\xi } \nonumber \\&-\,u^{2}\int _0^1 {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi } }''}{\varvec{qq}}}^\mathrm{T}\left( {\int _\xi ^1 {{\varvec{\varphi }'}^\mathrm{T}{{{\varvec{\varphi } }''}}\mathrm{d}\xi } } \right) \mathrm{d}\xi } \nonumber \\&-\, \int _0^1 {\varvec{\varphi }}^\mathrm{T}{{{{\varvec{\varphi }} }''}{\varvec{qq}}}^\mathrm{T}\left( \int _\xi ^1 {{{\varvec{\varphi } }''}}^\mathrm{T}{{{{{\varvec{\varphi } }'''}}}\mathrm{d}\xi } \right) \mathrm{d}\xi \end{aligned}$$
(A.1)

It is noted that the nonlinear damping and stiffness matrices are as a function of \({{\varvec{q}}}\) and \({{\dot{{\varvec{q}}}}}\), which will bring a large number of integrals in each step and hence a high calculation consumption. Thus, in this work, we introduce the following algebraic relationship to solve this calculation problem [44]:

$$\begin{aligned} \left( {{{\varvec{Aqp}}}^{{{\varvec{T}}}}{{\varvec{B}}}} \right) _{ij}= & {} \sum _k \sum _l A_{ik} q_k p_l B_{lj} \nonumber \\= & {} \sum _k {\sum _l {q_k \left( {{\varvec{A}}} \right) _{ik} \left( {{\varvec{B}}} \right) _{lj} p_l } } \nonumber \\= & {} {{\varvec{q}}}^\mathrm{T}{{\varvec{C}}}^{ij}{{\varvec{p}}} \end{aligned}$$
(A.2)

where matrix \({{\varvec{C}}}^{ij}\) is built as follows:

$$\begin{aligned} {{\varvec{C}}}^{ij}={{\varvec{A}}}_i ^\mathrm{T}\left( {{{\varvec{B}}}^\mathrm{T}} \right) _j \end{aligned}$$
(A.3)

in which \({{\varvec{A}}}\) and \({{\varvec{B}}}\) are two arbitrary square matrices, \({{\varvec{q}}}\) and \({{\varvec{p}}}\) are two arbitrary array. Equations. (A.2) and (A.3) indicate that the products of form \({{\varvec{Aqp}}}^{{{\varvec{T}}}}{{\varvec{B}}}\) are matrices whose components can be written in the form of \({{\varvec{q}}}^\mathrm{T}{{\varvec{C}}}^{ij}{{\varvec{p}}}\). \({{\varvec{C}}}^{ij}\) is a matrix as the product of the transposition of row i of matrix \({{\varvec{A}}}\) and row j of matrix \({{\varvec{B}}}^\mathrm{T}\).

Using Eqs. (A.2) and (A.3), we can get the components of nonlinear damping and stiffness matrices in Eq. (A.1) as follows:

$$\begin{aligned}&\left( {{{\varvec{C}}}_{{\varvec{N}}} } \right) _{ij} ={{\varvec{q}}}^\mathrm{T}\left( {{{\varvec{C}}}_{{{\varvec{N1}}}} ^{ij}-{{\varvec{C}}}_{{{\varvec{N3}}}} ^{ij}} \right) {{\dot{{\varvec{q}}}}}\nonumber \\&\quad +\,{{\varvec{q}}}^\mathrm{T}\left[ {2u\sqrt{\beta }\left( {-{{\varvec{C}}}_{{{\varvec{N2}}}} ^{ij}+{{\varvec{C}}}_{{{\varvec{N4}}}} ^{ij}-{{\varvec{C}}}_{{{\varvec{N5}}}} ^{ij}} \right) } \right] {{\varvec{q}}} \nonumber \\&\left( {{{\varvec{K}}}_{{\varvec{N}}} } \right) _{ij} ={{\dot{{\varvec{q}}}}}^\mathrm{T}2u\sqrt{\beta }{{\varvec{K}}}_{{{\varvec{N1}}}} ^{ij}{{\varvec{q}}}\nonumber \\&\quad +\,{{\varvec{q}}}^\mathrm{T}\left( {{\begin{array}{l} {{u}^{2}{{\varvec{K}}}_{{{\varvec{N2}}}} ^{ij}+3{{\varvec{K}}}_{{{\varvec{N3}}}} ^{ij}+{{\varvec{K}}}_{{{\varvec{N4}}}} ^{ij}-{u}^{2}{{\varvec{K}}}_{{{\varvec{N5}}}} ^{ij}} \\ {+{{\varvec{K}}}_{{{\varvec{N6}}}} ^{ij}+{u}^{2}{{\varvec{K}}}_{{{\varvec{N7}}}} ^{ij}-{{\varvec{K}}}_{{{\varvec{N8}}}} ^{ij}-{u}^{2}{{\varvec{K}}}_{{{\varvec{N9}}}} ^{ij}-{{\varvec{K}}}_{{{\varvec{N1}}}0} ^{ij}} \\ \end{array} }} \right) {{\varvec{q}}}\nonumber \\ \end{aligned}$$
(A.4)

where

$$\begin{aligned} {{\varvec{C}}}_{{{\varvec{N1}}}} ^{ij}= & {} \int _0^1 \left( {{\varvec{\varphi }}^\mathrm{T}{\varvec{\varphi }'}} \right) _i ^\mathrm{T}\left( {\left( {\int _0^\xi {{\varvec{\varphi }'}^\mathrm{T}{\varvec{\varphi }'}\mathrm{d}\xi } } \right) ^{\mathrm {T}}} \right) _j \mathrm{d}\xi ,\nonumber \\ {{\varvec{C}}}_{{{\varvec{N2}}}}^{ij}= & {} \int _0^1 {\left( {{\varvec{\varphi }}^\mathrm{T}{\varvec{\varphi }'}} \right) _i ^\mathrm{T}\left( {\left( {\int _0^\xi {{\varvec{\varphi }'}^\mathrm{T}{{{\varvec{\varphi } }''}}\mathrm{d}\xi } } \right) ^{\mathrm {T}}} \right) _j \mathrm{d}\xi } \nonumber \\ {{\varvec{C}}}_{{{\varvec{N3}}}} ^{ij}= & {} \int _0^1 \left( {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi } }''}}} \right) _i ^\mathrm{T}\left( {\left( {\int _\xi ^1 {\int _0^\xi {{\varvec{\varphi }'}^\mathrm{T}{\varvec{\varphi }'}\mathrm{d}\xi } } \mathrm{d}\xi } \right) ^{\mathrm {T}}} \right) _j \mathrm{d}\xi ,\nonumber \\ {{\varvec{C}}}_{{{\varvec{N4}}}} ^{ij}= & {} \int _0^1 {\left( {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi } }''}}} \right) _i ^\mathrm{T}\left( {\left( {\int _\xi ^1 {\int _0^\xi {{\varvec{\varphi }'}^\mathrm{T}{{{\varvec{\varphi } }''}}\mathrm{d}\xi } } \mathrm{d}\xi } \right) ^{\mathrm {T}}} \right) _j \mathrm{d}\xi } \nonumber \\ {{\varvec{C}}}_{{{\varvec{N5}}}} ^{ij}= & {} \int _0^1 {\left( {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi } }''}}} \right) _i ^\mathrm{T}\left( {\left( {\int _\xi ^1 {{\varvec{\varphi }'}^\mathrm{T}{\varvec{\varphi }'}\mathrm{d}\xi } } \right) ^{\mathrm {T}}} \right) _j \mathrm{d}\xi } ,\nonumber \\ {{\varvec{K}}}_{{{\varvec{N1}}}} ^{ij}= & {} \int _0^1 \left( {{\varvec{\varphi }}^\mathrm{T}{\varvec{\varphi }'}} \right) _i ^\mathrm{T}\left( {\left( {{\varvec{\varphi }'}^\mathrm{T}{\varvec{\varphi }'}} \right) ^{\mathrm {T}}} \right) _j \mathrm{d}\xi \nonumber \\ {{\varvec{K}}}_{{{\varvec{N2}}}} ^{ij}= & {} \int _0^1 \left( {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi } }''}}} \right) _i ^\mathrm{T}\left( {\left( {{\varvec{\varphi }'}^\mathrm{T}{\varvec{\varphi }'}} \right) ^{\mathrm {T}}} \right) _j \mathrm{d}\xi ,\nonumber \\ {{\varvec{K}}}_{{{\varvec{N3}}}} ^{{{\varvec{ij}}}}= & {} \int _0^1 {\left( {{\varvec{\varphi }}^\mathrm{T}{\varvec{\varphi }'}} \right) _i ^\mathrm{T}\left( {\left( {{{{\varvec{\varphi } }''}}^\mathrm{T}{ {{{{\varvec{\varphi }} }'''}}}} \right) ^{\mathrm {T}}} \right) _j \mathrm{d}\xi } \nonumber \\ {{\varvec{K}}}_{{{\varvec{N}}}4} ^{ij}= & {} \int _0^1 \left( {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi } }''}}} \right) _i ^\mathrm{T}\left( {\left( {{{{\varvec{\varphi } }''}}^\mathrm{T}{{{\varvec{\varphi } }''}}} \right) ^{\mathrm {T}}} \right) _j \mathrm{d}\xi ,\nonumber \\ {{\varvec{K}}}_{{{\varvec{N}}}5} ^{ij}= & {} \int _0^1 \left( {{\varvec{\varphi }}^\mathrm{T}{\varvec{\varphi }'}} \right) _i ^\mathrm{T}\left( {\left( {\int _0^\xi {{\varvec{\varphi }'}^\mathrm{T}{{{{{\varvec{\varphi }} }'''}}}\mathrm{d}\xi } } \right) ^{\mathrm {T}}} \right) _j \mathrm{d}\xi \nonumber \\ {{\varvec{K}}}_{{{\varvec{N}}}6} ^{ij}= & {} \int _0^1 \left( {{\varvec{\varphi }}^\mathrm{T}{\varvec{\varphi }'}} \right) _i ^\mathrm{T}\left( {\left( {\int _0^\xi {{{{\varvec{\varphi } }''}}^\mathrm{T}{{{{{{\varvec{\varphi }} }''''}}}}\mathrm{d}\xi } } \right) ^{\mathrm {T}}} \right) _j \mathrm{d}\xi ,\nonumber \\ {{\varvec{K}}}_{{{\varvec{N}}}7} ^{ij}= & {} \int _0^1 {\left( {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi } }''}}} \right) _i ^\mathrm{T}\left( {\left( {\int _\xi ^1 {\int _0^\xi {{\varvec{\varphi }'}^\mathrm{T}{{{{{{\varvec{\varphi }}} }'''}}}\mathrm{d}\xi } } \mathrm{d}\xi } \right) ^{\mathrm {T}}} \right) _j \mathrm{d}\xi } \nonumber \\ {{\varvec{K}}}_{{{\varvec{N}}}8} ^{ij}= & {} \int _0^1 \left( {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi } }''}}} \right) _i ^\mathrm{T}\left( {\left( {\int _\xi ^1 {\int _0^\xi {{{{\varvec{\varphi } }''}}^\mathrm{T}{{{{{{\varvec{\varphi }} }''''}}}}\mathrm{d}\xi } } \mathrm{d}\xi } \right) ^{\mathrm {T}}} \right) _j \mathrm{d}\xi ,\nonumber \\ {{\varvec{K}}}_{{{\varvec{N}}}9} ^{ij}= & {} \int _0^1 {\left( {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi } }''}}} \right) _i ^\mathrm{T}\left( {\left( {\int _\xi ^1 {{\varvec{\varphi }'}^\mathrm{T}{{{\varvec{\varphi } }''}}\mathrm{d}\xi } } \right) ^{\mathrm {T}}} \right) _j \mathrm{d}\xi } \nonumber \\ {{\varvec{K}}}_{{{\varvec{N1}}}0} ^{ij}= & {} \int _0^1 \left( {{\varvec{\varphi }}^\mathrm{T}{{{\varvec{\varphi } }''}}} \right) _i ^\mathrm{T}\left( {\left( {\int _\xi ^1 {{{{\varvec{\varphi } }''}}^\mathrm{T}{ {{{{\varvec{\varphi }} }'''}}}\mathrm{d}\xi } } \right) ^{\mathrm {T}}} \right) _j \mathrm{d}\xi \end{aligned}$$
(A.5)

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Zhou, K., Xiong, F.R., Jiang, N.B. et al. Nonlinear vibration control of a cantilevered fluid-conveying pipe using the idea of nonlinear energy sink. Nonlinear Dyn 95, 1435–1456 (2019). https://doi.org/10.1007/s11071-018-4637-8

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